Kinematics modeling of the gear-based crank mechanism engine regardless of the compressions ratio variations

In this work, geometric and kinematic modeling of the gear-based crank mechanism engine (GBCM) was performed. To this aim, a mechanical approach based on projective computational methods and mechanism theory laws is applied, to which a parametric study has allowed a conclusion on the geometrical and kinematic behaviors of the moving links. The study concluded that the kinematic quantities at connecting rod head are one-half of those at the piston top head. The extrinsic behavior like the stroke of the connecting rod head is twice the crank radius and the piston kinematic are identical to the conventional engine with the same crankshaft ratio regardless of the compression ratio and any gear wheel radius. Hence, all the extrinsic kinematic properties of a classic crankshaft mechanism of the fixed compression ratio engine remain valid for a gear-based crank mechanism engine and can be used for dynamic calculation purposes.


Acceleration vector of an M point of the n-th link θ 3
Angular displacement of gear wheel 3 in variable compression ratio configuration R 3

Radius of gear wheel 3 in variable compression ratio configuration
Environmental problems are making vehicle manufacturers lessen their fuel emissions by enhancing engine efficiency through optimized thermal control and growing the compression ratio.In this context, Variable compression ratio engines (VCR) make a contribution, in which the very last function of the Top dead center TDC piston is adjusted to optimize the compression ratio to increase thermodynamic performance, particularly at part load, and finally reduce gasoline consumption and CO2 emissions.Several VCR configurations presently exist [1][2][3][4][5] , such as the gear-based crank mechanism engine (GBCM) with variable compression ratio (VCR_MCE-5) [6][7][8][9] .What makes the GBCM engine distinctive from the traditional constant compression ratio engine is the interposition of a managed equipment wheel between the piston and the crankshaft mechanism, which gives the advantage of a variable compression ratio.
In this work, we can talk about the results received via the GBCM engine.To evaluate the results of the proposed mechanism, we simulated kinematic parameters together with piston stroke, positions, velocities, and accelerations, which can be used for dynamic computational purposes followed in analytical computational codes for multi-body simulation (MBS) [10][11][12][13] .
The results confirm that kinematic parameters are obtained for the GBCM engine regardless of the value of the compression ratio i.e. regardless of the position of the control cylinder.
The aim is to kinematically simulate the VCR engine, the results of which are used to describe the kinematic behavior through a pre-established parameter setting.
Firstly, kinematic modeling and simulations were performed then dynamic analyses can be performed taking into account external forces.The analysis processes were carried out according to the general methodology described in Fig. 1.In the second section, we study the behavior laws of the VCR engine through assumptions and adequate parameterization to establish the solutions in terms of strokes, positions, speeds, and accelerations.Finally, we discuss the influence of the parameters on the kinematic behavior of the VCR engine to conclude the evolution of the kinematic steady-state quantities.

Analytical study
Like in conventional engines, the crankshaft of the GBCM engine transforms an oscillating rectilinear displacement into a continuous rotation.The piston stroke between the TDC and BDC is not affected by the CR setting (Fig. 2).The hydraulic valve 6 and its control rack 7 are kinematically fixed so that have useful mobility equal to one.
A planar geometric parameterization A, x, y, 0 is used.The piston and the cylinder link are assumed to be a sliding link and the axis of the connecting rod head and the axis of the crank is concurrent: y = AC/�AC�.

In this section, a geometric closure connecting A-B-C-J-D-A points leads to the kinematic laws as follows:
Input parameter y C 3 denotes the translation of the gear wheel 3 in C. Output parameter θ 1 denotes the crankshaft rotation 1 in A. With: (1)

Geometric method
This approach is based on projection and derivation: The piston velocity 4 is: With: The no-slip rolling condition of 3 with respect to 4 gives: This involves: Finally, it is shown by time differentiation that the velocity and acceleration of piston 4 at D are two times greater than those calculated in the connecting rod head 2 at C: By derivation of ( 6) we obtain the speed as follows: The acceleration ÿD 4 in D of piston 4 is expressed as a function of the angular position θ 1 and the rotational speed θ1 of the crankshaft 1 in a steady state case by:

Kinematic method
The kinematic approach employs the law of moments, the equi-projectivity of fields velocities, no-slip rolling, and the velocity motion composition between parts.Therefore, the relationship of velocities between the piston at D and the connecting rod head at C in relation to the structure's frame: The no-slip rolling condition in relation to the structure's frame: (5) www.nature.com/scientificreports/Yet we have: The no-slip rolling condition of 3 with respect to 4 gives: It can be deduced that: As the acceleration is a non-equi-projective field, it is deduced from the temporal derivation of the field velocity.

Results and discussion
As with the fixed compression ratio engine, the GBCM engine also has input-output relationships in position, speed, and acceleration between the rotation of the crank 1 and the vertical displacement of the piston 4.This mechanism also has only one useful mobility θ 1 .
Figures 3, 4, and 5 shows the alternative rectilinear position, velocity, and acceleration at points C and D respectively expressed as a function of the angle θ 1 of the VCR engine crankshaft with = 1/1.5 .Thus, the kin- ematic quantities at point C are one-half of those at point D. As a result, only half the mass of the connecting rod is taken into account when calculating the inertial forces, giving the connecting rod increased structural rigidity to resist buckling.To obtain a unit displacement of 1484/4 cm3 with a bore of 75 mm in diameter, it is necessary to travel a stroke of (2*L1 = 2*21 mm).With regard to semi-fast engines, the use of a corresponding lambda factor equal to 1/1.5 is often recommended.This choice is based on several technical considerations.It aims to reduce mechanical constraints, improve the durability of the engine and optimize its energy efficiency.This ratio between the stroke and the of the connecting rod makes it possible to achieve a balance between engine speed and torque, thus ensuring optimal performance in different usage conditions.Thus, the piston kinematics is strictly identical to that of a conventional engine with the same crankshaft ratio regardless of the compression ratio and any gear wheel radius.
The stroke of the connecting rod head 2 at C is twice the crank radius C = 2L 1 like the classic fixed com- pression ratio engine according to Figs. 3, 4, and 5 which show respectively the behavior of the reciprocating rectilinear position, velocity, and acceleration of the connecting rod heads 3 at point C expressed as a function of the angle θ 1 of the crankshaft with V = 1/1.5 in a steady-state fixed at N = 3000 rpm.

Figure 1 .
Figure 1.General bloc diagram to evaluate the kinematic behaviour.

Figure 2 .
Figure 2. A representation of the GBCM engine structure.

Figure 3 .
Figure 3. Position at C and D vs. angular displacement of the VCR engine with = 1/1.5 and N = 3000 rpm.

Figure 4 .
Figure 4. Speed at C and D vs. angular displacement of the VCR engine with = 1/1.5 and N = 3000 rpm.

Figure 5 .
Figure 5. Acceleration at C and D vs. angular displacement of the GBCM engine with = 1/1.5 and N = 3000 rpm.